Astron. Astrophys. 342, 809-822 (1999)

5. Comparison with the chemical model of Paper II

The chemical model of Paper II represents an improvement on previous single-point time-dependent chemical models of hot cores by basing the chemical evolution of the gas around a physical and structural model of the cloud deduced from molecular line observations (Heaton et al. 1989, 1993). The physical model divides the cloud into three spherical components: an ultracompact core of temperature 300 K and density H₂ molecules per cm³, a surrounding compact core with temperature falling as K and density H₂ molecules per cm³, all embedded within a large halo 3.5 pc in radius with temperature falling as K and density falling as H₂ molecules per cm³. The parameters of the physical model are summarised in Table 3.

Table 3. Physical parameters of the structural model for G34.26+0.15 developed by Heaton et al. (1989)

The chemical model uses the values derived from the physical model to evaluate the temperature and density at 22 radial points within the model cloud. The innermost 3 points overlap to some degree and the cloud model thus contains 19 concentric "shells" of constant temperature and density. The structural model predicts that the physical conditions within the ultracompact and compact cores change relatively little from centre to edge. The ultracompact and compact cores are thus represented by a single shell each. The halo, where the conditions from inner edge to outer edge are very different, is modelled by the remaining 17 shells. Within each shell the chemical evolution is followed by a reaction network containing 225 species and 2184 separate reactions. The reaction network includes evolution from atomic gas and grain evaporated species.

The choice of method used to compare the model predictions with the observations is limited by the critical assumptions of both the analysis of the observational data and the chemical model. We outline the major assumptions of the analysis procedure and the chemical model below. The comparison of the model to the observations is also explained, as are the main caveats.

The main assumptions in our analysis of the observational data is that the gas is in LTE, optically thin and characterised by a single (rotational) temperature. For molecules with a high critical density (e.g. CS) the density of the halo may not be sufficient to thermalise the transitions and the assumption of LTE may break down. We are reasonably confident that most detected transitions are optically thin, the lines (with the exception of CO and ¹³CO) do not exhibit self-absorption. The more optically thin isotopomers of certain lines (e.g. C³⁴S, ³⁴SO2, H¹³CN) are not detected down to the noise level, indicating that at most the optical depths are of the order 2-3. The results from the analysis yield lower limits to the column density for most molecules, with methanol being the only species for which a strict temperature and column density could be derived.

The structural model upon which the chemical model is based assumes a spherical geometry for the cloud which clearly is not the case in reality (Fig. 1). However we note that the structural model on which the chemical model is based (Heaton et al. 1993) reproduces the observed line profiles extremely well by assuming spherical geometry. The column densities of the core in Millar et al. 1997 are calculated by mutiplying the fractional abundances produced by the chemical model by the appropriate cloud densities generated by the physical model and integrating along the line of sight through the core. These column densities depend upon the cloud geometry, and as the integration extends through the full depth of the cloud the emission is implicity assumed to be optically thin. The evolution timescales are dependent upon the initial abundances of the model, which for the halo arbitrarily consist of atomic gas (see Millar et al. 1997 for further details). The timescales for the halo derived from various species should thus not be taken too literally but in the absence of a more rigorous model (perhaps combining the physical and chemical evolution of the cloud) at least the broad timescales of the halo can be determined.

Given these difficulties in interpretation we have decided to follow the simplest means of comparing the model predictions to the observations by contrasting the predicted and observed column densities. Although our simple method does not take account of possibly optically thick emission and limits to column density the approach undertaken is important. With the advent of hot core chemical models that couple the chemistry to the physical conditions throughout the cloud rather than just at a single point, the questions of beam-averaging by the telescope, radiative transfer and the physical evolution of the cloud become extremely relevant. We aim to take the current single-point chemical models a step further by adding the concept of cloud structure (Paper II) and the beam-averaging effect of the telescope (this work). Comparison of the model predictions with the observations points to further improvements that can be made in the modelling process (such as the addition of radiative transfer, see Sect. 7) and identifies the need for further observations to constrain the observed column densities and provide additional data for more rigorous modelling of the halo and core.

Paper II concentrated on the model predictions for the column density observed along a central line of sight through the cloud, i.e. a "point" column density observed at the centre of the cloud. Here we extend and adapt the model to predict the column densities observed at an offset position within the halo and compare these predicted column densities with our observed values. We have adopted two approaches: a single line of sight drawn through the cloud much like the method used in Paper II and a beam-averaged approach, weighting multiple lines of sight drawn through the cloud with a gaussian to approximate the sampling effect of the telescope beam.

The "pencil-beam" single line of sight approach is illustrative to compare with the extended beams used in the beam-averaged model and the observations. Previous chemical models of hot cores have all relied on the single-point approach. It is interesting to see just how the beam-averaging process affects a single point model, i.e. how effective the single-point models were at predicting the beam-averaged observations. In addition there is an important difference between these two approaches. The "pencil-beam" drawn through the model cloud corresponding to the position of the halo survey does not intersect the cores and the column densities predicted by this model are representative of the halo chemistry alone. The beam-averaged model on the other hand is comprised of multiple lines of sight, some of which do intersect the core gas. The beam-averaged model shows that the effect of the core chemistry may extend to influence our observations made of the halo. By comparing the two approaches we can test whether the model halo chemistry alone can reproduce the observed column densities, or whether the areas of the core covered by the fringes of the telescope beam still dominate the observed column densities. Each approach is described in more detail in the following two sections.

5.1. Single line of sight model

Our first method of interpreting the radial chemical model of G34.26+0.15 was to evaluate the column densities along a "pencil-beam" line of sight corresponding to the centre of the telescope beam of our halo spectral line survey, i.e. or 0.3 pc from the centre of the cloud. It is a useful approach to compare the "pencil-beam" model with the extended beam of the observations because all previous models of hot cores have been single point (and thus pencil-beam models). Comparing the two models with the extended beam of the observations allows us to assess how effective a single point model is at predicting the column densities.

A diagram of the model is shown in Fig. 4, indicating some of the shells and their depths along the line of sight, radii, fractional abundances and densities ( and respectively). The shell depths along the line of sight () are evaluated geometrically as shown in Fig. 4. The density of each shell is calculated from the structural model and the fractional abundances of each species are taken from the chemical model of Paper II. The product of the fractional abundance of each species and the density of the gas within the shell gives the number density of each species within each shell. The depth of each shell is then used to obtain the column density of each species by integrating along the line of sight. The full thickness of the cloud along the line of sight is taken into account, which implicitly assumes that the emission from the species considered is optically thin.

Fig. 4. a and b  A side view of the single line of sight model. Some of the 19 concentric shells used in the chemical model are indicated with the line of sight used for our spectral line survey of the halo. The parameters of each shell, radius (ri), depth (di), density () and fractional abundance (fi) are shown. b  A frontal view of the single line of sight model showing the locations of the cloud centre (the ultracompact core) and the beam centre of the halo spectral survey.

To contrast our observations with the model predictions we have used models with halo ages of 10⁴, 10⁵ and 10⁶ years. We have compared the ratios of column density predicted by the model and that observed in our survey; these ratios can be seen in Table 4. The best fit to the observations is given by the model with a halo age of 10⁶ years, although it can be seen that for many of the molecules seen in the survey (e.g. CN, CS, H₂CO and HCN) the model predicts a 10-300 times greater column density than is observed.

Table 4. The ratio of model column density to observed column density for each model calculated with the single line of sight approach. The ratio for CO is calculated with the observed column density of C¹⁷O assuming the ¹⁶O/¹⁷O ratio of 2400. Observed column densities are lower limits with the exception of CH₃OH.

Different molecules appear to predict different ages; i.e. the CO column density ratio is favoured by an older halo of 10⁶ years whilst the SO₂ column density ratio has a good fit to a halo of between 10⁴ and 10⁵ years with older halos possessing a much larger column density of SO₂ than is observed towards G34.26+0.15. CH₃OH is another molecule whose predicted column density favours a younger halo of between 10⁴ and 10⁵ years, as the column density of CH₃OH falls rapidly as the halo increases in age.

We must reconcile the apparently different ages predicted by the single line of sight model. The single line of sight model represents only the cold gas-phase chemistry of the halo. However, our observed column densities are averaged across the telescope beam which is of finite size unlike the "pencil-beam" single line of sight drawn through the cloud. The telescope beam samples an area of the cloud which will include both warmer gas closer towards the core and colder gas further away from the core. The grain mantle evaporation chemistry of the hot dense gas in the cores is significantly different from the cold gas-phase chemistry of the halo. The column densities of species picked up in the fringes of the telescope beam will thus affect the overall beam-averaged column density. There is some evidence for this in the observations, the linewidths of HCN and CS in the halo are similar to their linewidths in the core. The linewidths of the other species are much less (typically half or a third of the core linewidth) which suggests that they originate mainly from the cooler gas of the halo. In the next section we describe a multiple line of sight approach to interpreting the model which attempts to simulate the beam-averaging effect of the telescope.

5.2. Beam-averaged model

As described above the single line of sight method does not reproduce accurately the CO column densities observed in the halo. The halo column densities observed in this survey may also have a contribution from the gas close to the cores. Our observations were carried out with a beam and offset from the central position of Paper I. At the distance of G34.26 (3.1 kpc) this is equivalent to an HPBW of 0.2 pc and an offset of 0.3 pc (the radius of the compact core is 0.1 pc for comparison). For a gaussian beam of this HPBW the distance at which the beam falls to the 1 level is 0.25 pc. In a simple one dimensional beam we can see that, considering the 1 level as a "cutoff", there is still a 0.05 pc overlap with the compact core (which has a radius of 0.1 pc). The column densities seen in the compact core for some species are between 2-3 orders of magnitude greater than in the halo. It is plausible that the dense gas of the cores sampled in the edge of the beam contributes roughly the same amount (or possibly more) towards the beam-averaged column density as does the halo gas.

Pointing offsets during the observations may exacerbate the problem of core pickup. The estimated maximum offset during the observations was , if it is assumed that the offset was toward the core then the beam centre will move 0.08 pc closer toward the core and the overlap with the compact core becomes 0.13 pc. However it is also likely that the pointing offsets could move the beam centre as far away from the core, negating the overlap entirely. As mentioned in the previous section, there is some evidence for core pickup in the observed linewidths of certain species (HCN and CS). This is not the case for the remaining species which have narrower halo linewidths than core linewidths. Once the structure of the cloud is taken into account by the chemical model it is important to consider the spatial sampling of the telescope and so to simulate the beam-averaged column density that the JCMT "sees" we have extended the single line of sight technique to include beam-averaging.

We use a square grid of equally spaced points centred on the beam centre, extending to the one percent level of a gaussian function corresponding to the telescope beam, to give the locations of the lines of sight to be drawn through the cloud. We have experimented with different levels of sampling the model to approximate to the JCMT beam of HPBW at 345 GHz. A relatively coarse level of sampling was first used with a separation between grid points of 0.1 pc and a 7 7 grid. This proved to have reasonable model run times but did not adequately sample the core gas where the dimensions are 0.01-0.1 pc (i.e. the grid spacing). A grid spacing of 0.01 pc (53 53 grid points) was found to be a good compromise between run time and sampling level. Decreases in the grid spacing below 0.01 pc have little effect on the beam-averaged column densities. A schematic of the beam-averaged model is given in Fig. 5. The gaussian beam corresponding to the telescope beam is used to obtain the appropriate weight for each line of sight and the weighted lines of sight are summed over the beam to give beam averaged column densities.

Fig. 5. a and b A schematic of the beam-averaged model with side view a and front view b . Both views are not to scale. a  Some representative lines of sight and the gaussian beam used to weight the column density along each line of sight are indicated. Shell depths (di) are evaluated for each line of sight in the beam in a similar manner to the single line of sight model. b  Some of the points in the equally spaced grid used to approximate the telescope beam are shown to illustrate that the fringes of the beam sample core gas.

We have calculated the ratios for a combination of twelve models with 4 different core ages (, 10³, & 10⁴ years) and 3 different halo ages (10⁴, 10⁵ & 10⁶ years). Model fits to the observations for each halo age and the oldest and youngest core ages are shown in Table 5.

Table 5. The ratio of model column density to observed column density for each model calculated with the beam-averaged approach. We have also included the youngest and oldest core models to illustrate the effects of core evolution on the beam-averaged column density. The ratio for CO is calculated with the observed column density of C¹⁷O assuming the ¹⁶O/¹⁷O ratio of 2400. Observed column densities are lower limits with the exception of CH₃OH.

From the results displayed in Table 5 it can be seen that on the whole the beam-averaged model does not produce significantly different results to the "pencil-beam" model. The column densities of most species are roughly 6-7 greater than the pencil-beam model, attributable to the beam-averaged H₂ column density being greater than the "pencil-beam" H₂ column density by roughly 7. However some molecules have a difference much greater than 7 which is evidence that the warmer gas in the fringes of the beam does affect the beam-averaged column densities (at least in the model). The clearest example of this is methanol, which in the old halo of 10⁶ years has a model column density raised by a factor of 400 compared to the single line of sight model.

The evolution of the core gas has little effect as the column density ratios of most molecules remain unchanged over the timescales of the core evolution. The ratios of CH₃OH, SO₂ and HCN are modified from the single line of sight model prediction when core evolution is taken into account. The model column density CH₃OH falls by roughly a factor of two as the core ages from 3.2 years, the SO₂ column density rises sharply for core evolution with a halo of 10⁴ years and the HCN column density rises marginally over all the halo ages, with the largest increase at a halo age of 10⁶ years.

CO again predicts a halo age of 10⁵-10⁶ years, as do the molecules C₂H and H₂CO. Other molecules are over-estimated in the old halo by factors of 10-300. CH₃OH gives an uncertain prediction since its column density is affected by the core evolution, from the CH₃OH evidence alone any halo age is plausible. SO₂ is over-produced in the model cloud at late halo ages by a factor of 10-100, the best prediction of the cloud age using SO₂ is for a young halo of 10⁴ years with a core age of between 3.2 10³ and 10⁴ years. We will interpret these and the single line of sight results in the next section.

© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
helpdesk.link@springer.de